Linear Algebra

348 Chapter Five. Similarity1.3 Example The only matrix similar to the zero matrix is itself: P ZP −1 =P Z = Z. The only matrix similar to the identity matrix is itself: P IP −1 =P P −1 = I.Since matrix similarity is a special case of matrix equivalence, if two matricesare similar then they are equivalent. What about the converse: mustmatrix equivalent square matrices be similar? The answer is no. The priorexample shows that the similarity classes are different from the matrix equivalenceclasses, because the matrix equivalence class of the identity consists ofall nonsingular matrices of that size. Thus, for instance, these two are matrixequivalent but not similar.( ) ( )1 01 2T =S =0 10 3So some matrix equivalence classes split into two or more similarity classes —similarity gives a finer partition than does equivalence. This picture shows somematrix equivalence classes subdivided into similarity classes.AB. . .To understand the similarity relation we shall study the similarity classes.We approach this question in the same way that we’ve studied both the rowequivalence and matrix equivalence relations, by finding a canonical form forrepresentatives ∗ of the similarity classes, called Jordan form. With this canonicalform, we can decide if two matrices are similar by checking whether theyreduce to the same representative. We’ve also seen with both row equivalenceand matrix equivalence that a canonical form gives us insight into the ways inwhich members of the same class are alike (e.g., two identically-sized matricesare matrix equivalent if and only if they have the same rank).Exercises1.4 For( )1 3S =−2 −6T =( )0 0−11/2 −5P =(4)2−3 2check that T = P SP −1 .̌ 1.5 Example 1.3 shows that the only matrix similar to a zero matrix is itself andthat the only matrix similar to the identity is itself.(a) Show that the 1×1 matrix (2), also, is similar only to itself.(b) Is a matrix of the form cI for some scalar c similar only to itself?(c) Is a diagonal matrix similar only to itself?1.6 Show that these matrices are not similar.( 1 0 41 1 32 1 7) ( 1 0) 10 1 13 1 2∗ More information on representatives is in the appendix.